The new Pixar movie Lightyear explains it pretty well in layman's terms.

There are a million videos on YouTube that will do a better job explaining this than you're likely to get here. Some visual explanations will be useful.

I'm not sure we actually know the 'why'. We just can see and prove that it does.

The way I understand it (no way am I a great source) but it's all about perspective. For those on the water planet, time moved normally, but because they were closer to the gravitational center of the black hole, their "normal" was much slower than the ships. (Why? Not sure that we know exactly. The reason is the massive gravitational pull warping time and space itself).

So, for those on the water planet, if they were looking through a telescope, the guy on the ship would be moving at super speed.

For the guy in the ship looking down, the people/giant wave would be moving in super slow motion.

The part where it talks about the lady to first arrive on the water planet who is now dead, explains this better. She is on the planet, gets hit by a wave and dies. Her signal thing sends out a frequency which is picked up by Mathew and his crew. They go all the way across space to get to the water planet.

Think of timeline here. They probably first got the signal once the first lady arrived. And in that time, they traveled across space, just to have missed her dying by a slim fraction. Meaning, while she was dying, they were moving SUPER fast to reach the planet.

I'm hoping this helps a little. It's hard to grasp the full "why" because we don't know much about space/time continuum and how warping reality is done.

In two sentences: Acceleration allows you to trade how quickly you cover a distance with how quickly you cover a time interval. The higher the speed you accelerate yourself to, the more distance passes, but the less time passes.

This is somewhat common sense--the faster you drive your car, the less time it takes to get somewhere. Where it gets funky is that the universe has set it so that, for something going the speed of light, time never passes. Human common sense would dictate that, if you go an infinite speed, time never passes. The universe somehow set this maximum speed at "speed of light" instead of "infinite speed"--and makes it so that everything that sees light sees it traveling at the speed of light.

This is counter-intuitive. If one car is traveling at 20 mph and one travels at 30 mph, the first car sees the second traveling 10 mph relative to itself. If there's a sudden flash of light in the distance, both cars see the light going at the same speed. To make this work, the universe has to change how the two cars individually measure space and time.

This is where acceleration comes in. If two people are moving at different constant speeds, they'll experience space and time differently--but there's no way to tell who is right. As far as science and philosophy goes, they're both right. As soon as one of them accelerates, they're using energy to change how they experience space and time while still measuring the speed of light as constant the entire time.

That constant-speed-of-light thing is what messes up everything. If you're accelerating a car down the road, a speed gun in the car should measure all the stuff passing by as going faster and faster--NOT constant. The universe goes wonky to keep light speed constant.

If you flew away from Earth in a spaceship and looked at Earth while doing so, common-sense says that everything would appear to be going in slow-motion--since you're getting further and further away, light has to travel a longer and longer distance to reach you. Now keeping in mind that your rocket had to accelerate to get you out of Earth's gravity well, the universe's wonkiness steps in so that they also appear to be going in slow motion because you've changed how you are now experiencing space and time.

Edit: to answer the Interstellar question. Acceleration is caused by an imbalance of forces--things pushing and pulling on you. This includes experiencing gravity. If you're in a gravity well, congratulations--you're accelerating compared to the rest of the universe, and are actually aging slower than the rest of the universe. With Earth's gravity, it's not a huge effect. If you're around a black hole, it can be huge. The closer you get to a black hole, the more you feel the force of gravity, the more you're accelerating, the slower time passes for you. In the movie, they got closer to the black hole when they got to the planet. The guy on the ship was further from the black hole, so time passed faster for him. The actual numbers of 7 hours vs 23 years just comes from math.

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Light travels at 300,000 km/s. Always, forever, no matter who is observing it.

So imagine Alice is aiming a laser pointer into space. She fires a pulse of light. The photons in the pulse are traveling up at 300,000 km/s.

Now imagine Bob is flying by on a rocket traveling at 100,000 km/s in the same direction as the laser pulse. Every second he should observe the laser beam traveling away him by only 200,000 kilometers, right?

Nope. He still observes the pulse traveling 300,000 km away from him every second.

But if he’s traveling at 100,000 km/s away from Alice, how is Alice ALSO observing the beam traveling at 300,000 km/s?

The answer is that Bob and Alice are experiencing time differently. When Alice looks at Bob, Bob appears to be moving at 2/3 speed, whereas Bob sees Alice moving 3/2 faster than him. That way, light still travels at 300,000 km per second for both of them. It’s just that “one second” is longer for Alice than it is for Bob.

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